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Theorem chfacfscmulgsum 23158

Description: Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)

**Hypotheses**
Ref	Expression
chfacfisf.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
chfacfisf.b	⊢ 𝐵 = (Base‘𝐴)
chfacfisf.p	⊢ 𝑃 = (Poly₁‘𝑅)
chfacfisf.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
chfacfisf.r	⊢ × = (._r‘𝑌)
chfacfisf.s	⊢ − = (-_g‘𝑌)
chfacfisf.0	⊢ 0 = (0_g‘𝑌)
chfacfisf.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
chfacfisf.g	⊢ 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
chfacfscmulcl.x	⊢ 𝑋 = (var₁‘𝑅)
chfacfscmulcl.m	⊢ · = ( ·_𝑠 ‘𝑌)
chfacfscmulcl.e	⊢ ↑ = (._g‘(mulGrp‘𝑃))
chfacfscmulgsum.p	⊢ + = (+_g‘𝑌)

**Assertion**
Ref	Expression
chfacfscmulgsum	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Distinct variable groups: 𝐵,𝑛 𝑛,𝑀 𝑛,𝑁 𝑅,𝑛 𝑛,𝑌 𝑛,𝑏 𝑛,𝑠,𝐵 0 ,𝑛 𝐵,𝑖,𝑠 𝑖,𝐺 𝑖,𝑀 𝑖,𝑁 𝑅,𝑖 𝑖,𝑋 𝑖,𝑌 ↑ ,𝑖 · ,𝑏,𝑖 𝑇,𝑛 − ,𝑛 × ,𝑛 𝑖,𝑛 Allowed substitution hints: 𝐴(𝑖, 𝑛, 𝑠, 𝑏) 𝐵(𝑏) 𝑃(𝑖, 𝑛, 𝑠, 𝑏) + (𝑖, 𝑛, 𝑠, 𝑏) 𝑅(𝑠, 𝑏) 𝑇(𝑖, 𝑠, 𝑏) · (𝑛, 𝑠) × (𝑖, 𝑠, 𝑏) ↑ (𝑛, 𝑠, 𝑏) 𝐺(𝑛, 𝑠, 𝑏) 𝑀(𝑠, 𝑏) − (𝑖, 𝑠, 𝑏) 𝑁(𝑠, 𝑏) 𝑋(𝑛, 𝑠, 𝑏) 𝑌(𝑠, 𝑏) 0 (𝑖, 𝑠, 𝑏)

**Proof of Theorem chfacfscmulgsum**
Step	Hyp	Ref	Expression
1		eqid 2761	. . 3 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		chfacfisf.0	. . 3 ⊢ 0 = (0_g‘𝑌)
3		chfacfscmulgsum.p	. . 3 ⊢ + = (+_g‘𝑌)
4		crngring 20452	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	4	anim2i 629	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6	5	3adant3 1150	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
7		chfacfisf.p	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘𝑅)
8		chfacfisf.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
9	7, 8	pmatring 22990	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
10	6, 9	syl 18	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
11		ringcmn 20491	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
12	10, 11	syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ CMnd)
13	12	adantr 486	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ CMnd)
14		nn0ex 12593	. . . 4 ⊢ ℕ₀ ∈ V
15	14	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ℕ₀ ∈ V)
16		simpll 779	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
17		simplr 781	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))))
18		simpr 490	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℕ₀)
19	16, 17, 18	3jca 1146	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ ℕ₀))
20		chfacfisf.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
21		chfacfisf.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
22		chfacfisf.r	. . . . 5 ⊢ × = (._r‘𝑌)
23		chfacfisf.s	. . . . 5 ⊢ − = (-_g‘𝑌)
24		chfacfisf.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
25		chfacfisf.g	. . . . 5 ⊢ 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
26		chfacfscmulcl.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝑅)
27		chfacfscmulcl.m	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑌)
28		chfacfscmulcl.e	. . . . 5 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
29	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulcl 23155	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ ℕ₀) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
30	19, 29	syl 18	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
31	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulfsupp 23157	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) finSupp 0 )
32		nn0disj 13758	. . . 4 ⊢ ((0...(𝑠 + 1)) ∩ (ℤ_≥‘((𝑠 + 1) + 1))) = ∅
33	32	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0...(𝑠 + 1)) ∩ (ℤ_≥‘((𝑠 + 1) + 1))) = ∅)
34		nnnn0 12594	. . . . . 6 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ₀)
35		peano2nn0 12627	. . . . . 6 ⊢ (𝑠 ∈ ℕ₀ → (𝑠 + 1) ∈ ℕ₀)
36	34, 35	syl 18	. . . . 5 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ₀)
37		nn0split 13757	. . . . 5 ⊢ ((𝑠 + 1) ∈ ℕ₀ → ℕ₀ = ((0...(𝑠 + 1)) ∪ (ℤ_≥‘((𝑠 + 1) + 1))))
38	36, 37	syl 18	. . . 4 ⊢ (𝑠 ∈ ℕ → ℕ₀ = ((0...(𝑠 + 1)) ∪ (ℤ_≥‘((𝑠 + 1) + 1))))
39	38	ad2antrl 741	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ℕ₀ = ((0...(𝑠 + 1)) ∪ (ℤ_≥‘((𝑠 + 1) + 1))))
40	1, 2, 3, 13, 15, 30, 31, 33, 39	gsumsplit2 20123	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))))
41		simpll 779	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
42		simplr 781	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))))
43		nncn 12324	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
44		add1p1 12578	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℂ → ((𝑠 + 1) + 1) = (𝑠 + 2))
45	43, 44	syl 18	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ → ((𝑠 + 1) + 1) = (𝑠 + 2))
46	45	ad2antrl 741	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 + 1) + 1) = (𝑠 + 2))
47	46	fveq2d 6881	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (ℤ_≥‘((𝑠 + 1) + 1)) = (ℤ_≥‘(𝑠 + 2)))
48	47	eleq2d 2847	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↔ 𝑖 ∈ (ℤ_≥‘(𝑠 + 2))))
49	48	biimpa 482	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → 𝑖 ∈ (ℤ_≥‘(𝑠 + 2)))
50	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmul0 23156	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ (ℤ_≥‘(𝑠 + 2))) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )
51	41, 42, 49, 50	syl3anc 1398	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )
52	51	mpteq2dva 5198	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) = (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 ))
53	52	oveq2d 7428	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 )))
54	4, 9	sylan2 605	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
55		ringmnd 20450	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
56	54, 55	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Mnd)
57	56	3adant3 1150	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Mnd)
58		fvex 6890	. . . . . . . 8 ⊢ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V
59	57, 58	jctir 530	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑌 ∈ Mnd ∧ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V))
60	59	adantr 486	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 ∈ Mnd ∧ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V))
61	2	gsumz 19012	. . . . . 6 ⊢ ((𝑌 ∈ Mnd ∧ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
62	60, 61	syl 18	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
63	53, 62	eqtrd 2796	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = 0 )
64	63	oveq2d 7428	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + 0 ))
65		fzfid 14096	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0...(𝑠 + 1)) ∈ Fin)
66		elfznn0 13734	. . . . . . . 8 ⊢ (𝑖 ∈ (0...(𝑠 + 1)) → 𝑖 ∈ ℕ₀)
67	66, 19	sylan2 605	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ ℕ₀))
68	67, 29	syl 18	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
69	68	ralrimiva 3155	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑖 ∈ (0...(𝑠 + 1))((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
70	1, 13, 65, 69	gsummptcl 20161	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌))
71	1, 3, 2	mndrid 18925	. . . 4 ⊢ ((𝑌 ∈ Mnd ∧ (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌)) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + 0 ) = (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))
72	57, 70, 71	syl2an2r 698	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + 0 ) = (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))
73	64, 72	eqtrd 2796	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))
74	34	ad2antrl 741	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℕ₀)
75	1, 3, 13, 74, 68	gsummptfzsplit 20126	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))))
76		elfznn0 13734	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ₀)
77	76, 30	sylan2 605	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
78	1, 3, 13, 74, 77	gsummptfzsplitl 20127	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))))
79	57	adantr 486	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Mnd)
80		0nn0 12602	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
81	80	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ ℕ₀)
82	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulcl 23155	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 0 ∈ ℕ₀) → ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌))
83	81, 82	mpd3an3 1491	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌))
84		oveq1 7419	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑖 ↑ 𝑋) = (0 ↑ 𝑋))
85		fveq2 6877	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝐺‘𝑖) = (𝐺‘0))
86	84, 85	oveq12d 7430	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = ((0 ↑ 𝑋) · (𝐺‘0)))
87	1, 86	gsumsn 20148	. . . . . . 7 ⊢ ((𝑌 ∈ Mnd ∧ 0 ∈ ℕ₀ ∧ ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌)) → (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((0 ↑ 𝑋) · (𝐺‘0)))
88	79, 81, 83, 87	syl3anc 1398	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((0 ↑ 𝑋) · (𝐺‘0)))
89	88	oveq2d 7428	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))))
90	78, 89	eqtrd 2796	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))))
91		ovexd 7447	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ V)
92		1nn0 12603	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
93	92	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 1 ∈ ℕ₀)
94	74, 93	nn0addcld 12652	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ₀)
95	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulcl 23155	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ (𝑠 + 1) ∈ ℕ₀) → (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
96	94, 95	mpd3an3 1491	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
97		oveq1 7419	. . . . . . 7 ⊢ (𝑖 = (𝑠 + 1) → (𝑖 ↑ 𝑋) = ((𝑠 + 1) ↑ 𝑋))
98		fveq2 6877	. . . . . . 7 ⊢ (𝑖 = (𝑠 + 1) → (𝐺‘𝑖) = (𝐺‘(𝑠 + 1)))
99	97, 98	oveq12d 7430	. . . . . 6 ⊢ (𝑖 = (𝑠 + 1) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))
100	1, 99	gsumsn 20148	. . . . 5 ⊢ ((𝑌 ∈ Mnd ∧ (𝑠 + 1) ∈ V ∧ (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))
101	79, 91, 96, 100	syl3anc 1398	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))
102	90, 101	oveq12d 7430	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))))
103		fzfid 14096	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1...𝑠) ∈ Fin)
104		simpll 779	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
105		simplr 781	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))))
106		elfznn 13667	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
107	106	nnnn0d 12648	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ₀)
108	107	adantl 487	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ₀)
109	104, 105, 108, 29	syl3anc 1398	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
110	109	ralrimiva 3155	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑖 ∈ (1...𝑠)((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
111	1, 13, 103, 110	gsummptcl 20161	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌))
112	1, 3	mndass 18912	. . . . 5 ⊢ ((𝑌 ∈ Mnd ∧ ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌) ∧ ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))))
113	79, 111, 83, 96, 112	syl13anc 1399	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))))
114	106	nnne0d 12369	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ≠ 0)
115	114	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ 0)
116		neeq1 3018	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑖 → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
117	116	adantl 487	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
118	115, 117	mpbird 260	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ 0)
119		eqneqall 2967	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝑛 ≠ 0 → 0 = (𝑇‘(𝑏‘(𝑖 − 1)))))
120	118, 119	mpan9 516	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = (𝑇‘(𝑏‘(𝑖 − 1))))
121		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 𝑛 = 𝑖)
122		eqeq1 2765	. . . . . . . . . . . . . . . . 17 ⊢ (0 = 𝑛 → (0 = 𝑖 ↔ 𝑛 = 𝑖))
123	122	eqcoms 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → (0 = 𝑖 ↔ 𝑛 = 𝑖))
124	123	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (0 = 𝑖 ↔ 𝑛 = 𝑖))
125	121, 124	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = 𝑖)
126	125	fveq2d 6881	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑏‘0) = (𝑏‘𝑖))
127	126	fveq2d 6881	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑇‘(𝑏‘0)) = (𝑇‘(𝑏‘𝑖)))
128	127	oveq2d 7428	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))
129	120, 128	oveq12d 7430	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
130		elfz2 13627	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) ↔ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)))
131		zleltp1 12728	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
132	131	ancoms 464	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
133	132	3adant1 1148	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
134	133	biimpcd 252	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ≤ 𝑠 → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
135	134	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠) → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
136	135	impcom 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → 𝑖 < (𝑠 + 1))
137	136	orcd 887	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖))
138		zre 12678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℤ → 𝑠 ∈ ℝ)
139		1red 11290	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℤ → 1 ∈ ℝ)
140	138, 139	readdcld 11319	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 ∈ ℤ → (𝑠 + 1) ∈ ℝ)
141		zre 12678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
142	140, 141	anim12ci 626	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
143	142	3adant1 1148	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
144		lttri2 11373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
145	143, 144	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
146	145	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
147	137, 146	mpbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → 𝑖 ≠ (𝑠 + 1))
148	130, 147	sylbi 220	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ≠ (𝑠 + 1))
149	148	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ (𝑠 + 1))
150		neeq1 3018	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
151	150	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
152	149, 151	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ (𝑠 + 1))
153	152	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ (𝑠 + 1))
154	153	neneqd 2961	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = (𝑠 + 1))
155	154	pm2.21d 122	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → (𝑛 = (𝑠 + 1) → (𝑇‘(𝑏‘𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
156	155	imp 412	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏‘𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
157	106	nnred 12331	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℝ)
158		eleq1w 2844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑖 → (𝑛 ∈ ℝ ↔ 𝑖 ∈ ℝ))
159	157, 158	syl5ibrcom 250	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) → (𝑛 = 𝑖 → 𝑛 ∈ ℝ))
160	159	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑛 = 𝑖 → 𝑛 ∈ ℝ))
161	160	imp 412	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ∈ ℝ)
162	74	nn0red 12649	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℝ)
163	162	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑠 ∈ ℝ)
164		1red 11290	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 1 ∈ ℝ)
165	163, 164	readdcld 11319	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑠 + 1) ∈ ℝ)
166	130, 136	sylbi 220	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 < (𝑠 + 1))
167	166	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 < (𝑠 + 1))
168		breq1 5106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑖 → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
169	168	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
170	167, 169	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 < (𝑠 + 1))
171	161, 165, 170	ltnsymd 11440	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ¬ (𝑠 + 1) < 𝑛)
172	171	pm2.21d 122	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ((𝑠 + 1) < 𝑛 → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
173	172	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → ((𝑠 + 1) < 𝑛 → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
174	173	imp 412	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
175		simp-4r 796	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 = 𝑖)
176	175	fvoveq1d 7434	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘(𝑛 − 1)) = (𝑏‘(𝑖 − 1)))
177	176	fveq2d 6881	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘(𝑛 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
178	175	fveq2d 6881	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘𝑛) = (𝑏‘𝑖))
179	178	fveq2d 6881	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘𝑛)) = (𝑇‘(𝑏‘𝑖)))
180	179	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))
181	177, 180	oveq12d 7430	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
182	174, 181	ifeqda 4519	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
183	156, 182	ifeqda 4519	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
184	129, 183	ifeqda 4519	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
185		ovexd 7447	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) ∈ V)
186	25, 184, 108, 185	fvmptd2 6994	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝐺‘𝑖) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
187	186	oveq2d 7428	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
188	187	mpteq2dva 5198	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))
189	188	oveq2d 7428	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
190		nn0p1gt0 12616	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ℕ₀ → 0 < (𝑠 + 1))
191		0red 11292	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ₀ → 0 ∈ ℝ)
192		ltne 11388	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
193	191, 192	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ₀ ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
194		neeq1 3018	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑠 + 1) → (𝑛 ≠ 0 ↔ (𝑠 + 1) ≠ 0))
195	193, 194	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ₀ ∧ 0 < (𝑠 + 1)) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
196	34, 190, 195	syl2anc2 597	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
197	196	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
198	197	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → 𝑛 ≠ 0)
199		eqneqall 2967	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → (𝑛 ≠ 0 → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏‘𝑠))))
200	198, 199	mpan9 516	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏‘𝑠)))
201		iftrue 4488	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑠 + 1) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = (𝑇‘(𝑏‘𝑠)))
202	201	ad2antlr 740	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = (𝑇‘(𝑏‘𝑠)))
203	200, 202	ifeqda 4519	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = (𝑇‘(𝑏‘𝑠)))
204	74, 35	syl 18	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ₀)
205		fvexd 6892	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ V)
206	25, 203, 204, 205	fvmptd2 6994	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐺‘(𝑠 + 1)) = (𝑇‘(𝑏‘𝑠)))
207	206	oveq2d 7428	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) = (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))))
208	4	3ad2ant2 1152	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
209		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑃) = (Base‘𝑃)
210	26, 7, 209	vr1cl 22515	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
211	208, 210	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃))
212		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
213	212, 209	mgpbas 20345	. . . . . . . . . . . . 13 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
214		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (1_r‘𝑃) = (1_r‘𝑃)
215	212, 214	ringidval 20389	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑃) = (0_g‘(mulGrp‘𝑃))
216	213, 215, 28	mulg0 19264	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (Base‘𝑃) → (0 ↑ 𝑋) = (1_r‘𝑃))
217	211, 216	syl 18	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0 ↑ 𝑋) = (1_r‘𝑃))
218	7	ply1crng 22496	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
219	218	anim2i 629	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
220	219	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
221	8	matsca2 22715	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
222	220, 221	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌))
223	222	fveq2d 6881	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (1_r‘𝑃) = (1_r‘(Scalar‘𝑌)))
224	217, 223	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0 ↑ 𝑋) = (1_r‘(Scalar‘𝑌)))
225	224	adantr 486	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0 ↑ 𝑋) = (1_r‘(Scalar‘𝑌)))
226	225	oveq1d 7427	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ 𝑋) · (𝐺‘0)) = ((1_r‘(Scalar‘𝑌)) · (𝐺‘0)))
227	7, 8	pmatlmod 22991	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod)
228	4, 227	sylan2 605	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
229	228	3adant3 1150	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod)
230	20, 21, 7, 8, 22, 23, 2, 24, 25	chfacfisf 23152	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(Base‘𝑌))
231	4, 230	syl3anl2 1440	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(Base‘𝑌))
232	231, 81	ffvelcdmd 7077	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐺‘0) ∈ (Base‘𝑌))
233		eqid 2761	. . . . . . . . . 10 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
234		eqid 2761	. . . . . . . . . 10 ⊢ (1_r‘(Scalar‘𝑌)) = (1_r‘(Scalar‘𝑌))
235	1, 233, 27, 234	lmodvs1 21145	. . . . . . . . 9 ⊢ ((𝑌 ∈ LMod ∧ (𝐺‘0) ∈ (Base‘𝑌)) → ((1_r‘(Scalar‘𝑌)) · (𝐺‘0)) = (𝐺‘0))
236	229, 232, 235	syl2an2r 698	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((1_r‘(Scalar‘𝑌)) · (𝐺‘0)) = (𝐺‘0))
237		iftrue 4488	. . . . . . . . 9 ⊢ (𝑛 = 0 → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
238		ovexd 7447	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ V)
239	25, 237, 81, 238	fvmptd3 7009	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐺‘0) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
240	226, 236, 239	3eqtrd 2800	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ 𝑋) · (𝐺‘0)) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
241	207, 240	oveq12d 7430	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 ↑ 𝑋) · (𝐺‘0))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
242	1, 3	cmncom 19992	. . . . . . 7 ⊢ ((𝑌 ∈ CMnd ∧ ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 ↑ 𝑋) · (𝐺‘0))))
243	13, 83, 96, 242	syl3anc 1398	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 ↑ 𝑋) · (𝐺‘0))))
244		ringgrp 20444	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
245	10, 244	syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp)
246	245	adantr 486	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Grp)
247	207, 96	eqeltrrd 2862	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌))
248	10	adantr 486	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Ring)
249	24, 20, 21, 7, 8	mat2pmatbas 23024	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
250	4, 249	syl3an2 1182	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
251	250	adantr 486	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
252		simpl1 1210	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑁 ∈ Fin)
253	208	adantr 486	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
254		elmapi 8853	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
255	254	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
256	255	adantl 487	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
257		0elfz 13738	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ₀ → 0 ∈ (0...𝑠))
258	34, 257	syl 18	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
259	258	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ (0...𝑠))
260	256, 259	ffvelcdmd 7077	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
261	24, 20, 21, 7, 8	mat2pmatbas 23024	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
262	252, 253, 260, 261	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
263	1, 22	ringcl 20457	. . . . . . . 8 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
264	248, 251, 262, 263	syl3anc 1398	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
265	1, 2, 23, 3	grpsubadd0sub 19217	. . . . . . 7 ⊢ ((𝑌 ∈ Grp ∧ (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌) ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
266	246, 247, 264, 265	syl3anc 1398	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
267	241, 243, 266	3eqtr4d 2806	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
268	189, 267	oveq12d 7430	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
269	113, 268	eqtrd 2796	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
270	75, 102, 269	3eqtrd 2800	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
271	40, 73, 270	3eqtrd 2800	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 Vcvv 3451 ∪ cun 3897 ∩ cin 3898 ∅c0 4279 ifcif 4482 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 ↑_m cmap 8831 Fincfn 8957 ℂcc 11179 ℝcr 11180 0cc0 11181 1c1 11182 + caddc 11184 < clt 11324 ≤ cle 11325 − cmin 11522 ℕcn 12316 2c2 12378 ℕ₀cn0 12587 ℤcz 12674 ℤ_≥cuz 12946 ...cfz 13620 Basecbs 17367 +_gcplusg 17408 ._rcmulr 17409 Scalarcsca 17411 ·_𝑠 cvsca 17412 0_gc0g 17590 Σ_g cgsu 17591 Mndcmnd 18903 Grpcgrp 19124 -_gcsg 19126 ._gcmg 19257 CMndccmn 19974 mulGrpcmgp 20340 1_rcur 20387 Ringcrg 20439 CRingccrg 20440 LModclmod 21115 var₁cv1 22474 Poly₁cpl1 22475 Mat cmat 22702 matToPolyMat cmat2pmat 23002

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-ofr 7683 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-sup 9418 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-rp 13102 df-fz 13621 df-fzo 13769 df-seq 14125 df-hash 14455 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-hom 17432 df-cco 17433 df-0g 17592 df-gsum 17593 df-prds 17598 df-pws 17600 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-mhm 18958 df-submnd 18959 df-grp 19127 df-minusg 19128 df-sbg 19129 df-mulg 19258 df-subg 19313 df-ghm 19408 df-cntz 19511 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-cring 20442 df-subrng 20778 df-subrg 20802 df-lmod 21117 df-lss 21187 df-sra 21428 df-rgmod 21429 df-dsmm 22018 df-frlm 22033 df-ascl 22143 df-psr 22197 df-mvr 22198 df-mpl 22199 df-opsr 22201 df-psr1 22478 df-vr1 22479 df-ply1 22480 df-mamu 22686 df-mat 22703 df-mat2pmat 23005

This theorem is used by: cpmadugsum 23176

W3C validator